The Planar Turán Number of {K4,C5} and {K4,C6}

Let H be a set of graphs. The planar Tur & aacute;n number, exP(n,H), is the maximum number of edges in an n-vertex planar graph which does not contain any member of H as a subgraph. When H={H} has only one element, we usually write exP(n,H) instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both exP(n,C-5) and exP(n,K4). Later on, we obtained sharper bound for exP(n,{K4,C7}). In this paper, we give upper bounds of exP(n, {K-4,C-5}) <= 15/7(n-2) and exP(n,{K-4,C-6}) <= 7/3(n-2). We also give constructions which show the bounds are tight for infinitely many graphs.